Geometric and Topological Data Analysis: Big Picture, Lecture notes of Probability and Statistics

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Geometric and Topological Data

Analysis

Yen-Chi Chen

Department of Statistics

University of Washington

Geometric and Topological Data Analysis: Big Picture

Geometric and Topological Data Analysis: Big Picture

The data can be viewed as

X

, · · · , X

n

∼ p,

p is a probability density function.

Scientists are interested in geometric

or topological features of p.

Geometric and Topological Data Analysis: Big Picture

The data can be viewed as

X

, · · · , X

n

∼ p,

p is a probability density function.

Scientists are interested in geometric

or topological features of p.

Geometric and Topological Data Analysis: Big Picture

The data can be viewed as

X

, · · · , X

n

∼ p,

p is a probability density function.

Scientists are interested in geometric

or topological features of p.

Geometric and Topological Data Analysis: Big Picture

The data can be viewed as

X

, · · · , X

n

∼ p,

p is a probability density function.

Scientists are interested in geometric

or topological features of p.

The Classical Approach

◦ In all the above examples, how we estimate the

geometric/topological structures is based on plug-in estimates

from the kernel density estimator (KDE).

◦ Namely, we estimate the probability density function first and

then convert it into an estimator of the corresponding structure.

The Classical Approach

◦ In all the above examples, how we estimate the

geometric/topological structures is based on plug-in estimates

from the kernel density estimator (KDE).

◦ Namely, we estimate the probability density function first and

then convert it into an estimator of the corresponding structure.

◦ But this idea may fail.

Failure of KDE in Analyzing Data

Density Ranking: Introduction

◦ The KDE cannot detect intricate structures inside the GPS data.

◦ But the density ranking works!

Density Ranking: Introduction

◦ The KDE cannot detect intricate structures inside the GPS data.

◦ But the density ranking works!

◦ This comes from the fact that the underlying probability density

function (PDF) does not exist!

◦ Namely, our probability distribution function is a singular

measure.

Density

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Definition of Density Ranking - 3

◦ The density ranking is a transformed quantity from the KDE.

◦ Instead of using the density value, we focus on the ranking of it.